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Gravitation

Chapter: Gravitation

Gravitation or gravity is defined as a natural phenomenon in which all physical bodies attract each other. Gravity gives weight to physical objects and makes them to fall towards the ground when dropped.

Universal Law of Gravitation:

Newton’s law of gravitation states that any two objects in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Mathematically:

Let two objects A and B of masses m1 and m2 lie at a distance 'd' from each other as shown the figure below:

Let the force of attraction between two objects be F. According to the universal law of gravitation, the force between two objects is directly proportional to the product of their masses. That is,

---------------------(i)

And the force between two objects is inversely proportional to the square of the distance between them, that is:

---------------------(ii)

Combining the above equations we get-

or

'G' is the constant of proportionality and is called the universal gravitation constant. The SI unit of G is N m2 kg–2.
The value of 'G' was found out by Henry Cavendish (1731 – 1810) by using a sensitive balance. The accepted value of G is 6.673 × 10–11 N m2 kg–2.

Importance of the Universal Law of Gravitation

The universal law of gravitation successfully explained several phenomena like:

i. The force that binds things to the Earth.
ii. The revolution of the Moon around the Earth.
iii. The revolution of planets around the Sun.
iv. The tides due to the moon and the Sun.

Free Fall

We know that the earth attracts objects towards it.  This happens due to the gravitational force. Whenever objects fall towards the earth under this gravitational force alone, we conclude that the objects are in free fall. 
While falling there is generally no change in the direction of motion of the objects. But there will be a change in the magnitude of the velocity, due to the earth’s attraction. Any change in velocity also includes acceleration. Whenever an object falls towards the earth, acceleration is involved. This acceleration due to the Earth’s gravitational force. Therefore, this acceleration is called acceleration due to gravity. It is denoted by 'g'. The unit of 'g' is the same as that of acceleration, which is ms-2.

To calculate the value of 'g'

To calculate the value of g, we should put the values of G, M and R in the below equation, namely,

Universal gravitational constant, G = 6.7 × 10–11 N m2 kg-2,

Mass of the Earth,                    M = 6 × 1024 kg, and

Radius of the Earth,                  R = 6.4 × 106 m.

Substituting all the above values in this equation we get:

                                            g=9.8 m s-2

Motion of Objects Under the Influence of Gravitational Force of the Earth:

We learnt that every object experiences acceleration during free fall. This acceleration is independent of its mass. This means that all objects hollow or solid, big or small should fall at the same rate. According to a story, Galileo dropped different objects from the top of the Leaning Tower of Pisa in Italy to prove the same. As g is constant near the earth, all the equations for the uniformly accelerated motion of objects become valid with acceleration a replaced by 'g'.

The equations are:

Where 'u' and 'v' are the initial and final velocities and 's' is the distance covered in time 't'.

In applying these equations, we will consider acceleration, A to be positive when it is in the direction of the velocity, that is, in the direction of motion. The acceleration, A will be considered as negative when it opposes the motion.

Mass

Mass is defined as the property of a physical body which determines the body's resistance to being accelerated by a force and the strength of its mutual gravitational attraction with other bodies. Simply, it is quantity of matter contained in a body. The SI unit of mass is the kilogram (kg).

Weight

The weight of an object is usually taken to be the force on the object due to gravity. The SI unit of weight is Newton (N).

Weight of an object on the moon

We have learnt that the weight of an object on the earth is the force with which the earth attracts the object. In the same way, the weight of an object on the moon is the force with which the moon attracts that object. The mass of the moon is less than that of the earth. Due to this, the moon exerts lesser force of attraction on objects.

Weight of the object on the moon = (1/6) x its weight on the earth

Thrust and Pressure

Let us take two situations as examples and get better understanding of thrust and pressure.

Situation 1:
You think of fixing a poster on the bulletin board. So, you will take some drawing pins and press them with your thumb. Then the poster gets fixed on the board. Here you are applying force on the surface area of the head of the pin. This force is directed perpendicular to the surface area of the board. This force acts on a smaller area at the tip of the pin.

Situation 2:
When you go to some beach, you always like to play with the sand. When you stand on the sand, you must have observed your feet go deep into the sand. Now, lie down on the sand. And you will notice that your body will not go that deep in the sand. In both cases, the force exerted on the sand is the weight of your body.

Weight is the force acting vertically downwards always. Here the force is acting perpendicular to the surface of the sand. The force acting on any object perpendicular to the surface is called thrust. When you stand on loose sand, the force, which is the weight of your body, is acting on an area which is equal to the area of your feet. When you lie down, the same force acts on an area equal to the contact area of your whole body and it is larger than the area of your feet. Thus, the effects of forces of the same magnitude on different areas are different. In the above cases, though thrust is the same the effects are different. Therefore the effect of thrust depends on the area on which it acts. Thus we can conclude that the effect of thrust on sand is more while standing than while lying.

The thrust per unit area is called pressure.
 

Substituting the SI unit of thrust and area in the above equation, we get the SI unit of pressure as N/m2 or N m-2.

Pressure in Fluids

All liquids and gases are classified as fluids. A solid exerts pressure on a surface due to its weight. In the same way, fluids have weight, and they also exert pressure on the base and walls of the container in which they are enclosed. Pressure exerted in any confined mass of fluid is transmitted undiminished in all directions.

Buoyancy

Have you ever felt lighter when swimming in a pool? Have you ever drawn water from a well and felt that the bucket of water is heavier when it is out of the water? Have you ever wondered why a ship made of iron and steel does not sink in sea water, but while the same amount of iron and steel in the form of a sheet would sink? These questions can be answered by considering buoyancy.
The force due to the gravitational attraction of the earth acts on any object in the downward direction. So that object is pulled downwards. But the water exerts an upward force on the object. Thus, the object is pushed upwards. We already have learnt that the weight of an object is defined as the force due to gravitational attraction of the earth. When any object is immersed, we can observe that the upward force exerted by the water on the bottle is greater than its weight. Hence it rises up when released. The upward force on the object due to water must be balanced to keep the object completely immersed in water. This can be obtained by an externally applied force acting downwards. We should see that this force must at least be equal to the difference between the upward force and the weight of the bottle. The upward force exerted by the water on the object is known as up thrust or buoyant force. The magnitude of this buoyant force depends on the density of the fluid.

Why do objects Float or Sink When Placed on the Surface of Water?

Any object floats in water if the weight of the object is less than the weight of water displaced and it sinks if vice versa.
Let us consider an example in which we will study phenomena of a boat or a ship. A ship will float when the weight of the water it displaces equals the weight of the ship. This is termed as the Archimedes Principle as water exerts pressure on the hull of the ship with a net force pushing it upwards. If the area of the hull is large enough, then at some point the weight of the ship will the force that is pushing up the ship balance itself. This is the reason why heavy solid objects sink, but it floats if we keep on increasing the surface area to a larger area.
 

Archimedes Principle:

When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it. Archimedes principle has  many applications. It is used in designing ships and submarines. Lactometers, which are used to determine the purity of a sample of milk and hydrometers used for determining density of liquids, are based on this principle.

Relative Density

We already know that the density of a substance is defined as mass of a unit volume. The unit of density is kilogram per meter cube (kg m-3). The density of a given substance, under specified conditions, remains the same. Therefore the density of a substance is one of its characteristic properties. It is different for different substances. For example, the density of gold is 19300 kg m-3 while that of water is 1000 kg m-3. The density of a given sample of a substance can help us to determine its purity. It is often convenient to express density of a substance in comparison with that of water. The relative density of a substance is the ratio of its density to that of water:

Since the relative density is a ratio of similar quantities. So, it has no unit.